In the sphere of generalized mad families, to which this work belongs, the most relevant precursors are results on the cardinal invariants \(\mathfrak {a}_e\) , \(\mathfrak {a}_p\) , and \(\mathfrak {a}_g\) , the minimum cardinalities of a maximal eventually different family of functions on \(\mathbb {N}\) , a maximal almost disjoint family of permutations of \(\mathbb {N}\) , and a maximal cofinitary group of permutations of \(\mathbb {N}\) , respectively. A characterization of \(\mathrm {non}(\mathcal {M})\)  (Theorem REF  below) implies that \(\mathrm {non}(\mathcal {M})\le \mathfrak {a}_e\) , and moreover, \(\diamondsuit (\mathbb {N}^\mathbb {N},=^\infty )\)  implies \(\mathfrak {a}_e=\aleph _1\)  [1]}. Likewise, Brendle, Spinas and Zhang [2]} showed that \(\mathrm {non}(\mathcal {M})\le \mathfrak {a}_p,\mathfrak {a}_g\) , and Kastermans and Zhang [3]} showed that \(\diamondsuit (\mathbb {N}^\mathbb {N},=^\infty )\)  implies \(\mathfrak {a}_p=\mathfrak {a}_g=\aleph _1\) .
